91Ó°ÊÓ

Justify the terms "open ball" and "closed ball" by proving that (a) any open ball is an open set, (b) any closed ball is a closed set.

Show that the Cauchy-Schwarz inequality (11) implies $$ \left(\left|\xi_{1}\right|+\cdots+\left|\xi_{n}\right|\right)^{2} \leqq n\left(\left|\xi_{1}\right|^{2}+\cdots+\left|\xi_{n}\right|^{2}\right) $$

Show that \(C[0,1]\) and \(C[a, b]\) are isometric.

Show that a mapping \(T: X \longrightarrow Y\) is continuous if and only if the inverse image of any closed set \(M \subset Y\) is a closed set in \(X\).